Glauberman correspondence

Definition

Suppose $S$ is a finite solvable group acting on a finite group $G$. Assume that the orders of $S$ and $G$ are relatively prime. Then, the Glauberman correspondence is a natural bijection between the $S$-invariant irreducible linear characters of $G$, and the $S$-invariant irreducible linear characters of $C_{G}(S)$ (the fixed points in $G$ under the $S$-action).